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note
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We obtain the LIPs by diagonalizing the Hamiltonian terms V- S (t) C [see Eq.]. If the coupling between V2 and V3 is zero, the coupling matrix is Clc = μ12 1〉 〈2 + μ13 1〉 〈3 +c.c., where j〉 with j=1,2,3, refers to the electronic wave function, and c.c. stands for the complex conjugate terms. If we change the phases of the wave functions as ψ1′ = ψ1, ψ2′ =- ψ2, and ψ3′ =- ψ3, in the new representation the coupling matrix is C lc ′ =- Clc, so that the Hamiltonian in the absence of the pump pulse (which is treated as a perturbation) would look like H lc ′ =T+V+E (t) C. That is, a change in the overall phase of the control pulse (ES →- ES) amounts to a change in the phase of the wave functions, which cannot generate any observable difference in the dynamics. However, if we use the full coupling matrix C, and we apply the previous change of representation, we obtain H′ =T+V+ ES (t) (μ12 1〉 〈2 + μ13 1〉 〈3-μ23 2〉 〈3 +c.c.), which is not equivalent to a change of phase of the control pulse. In fact, it can be shown that there is no choice of representation such that H′ = R HR=T+V+ ES (t) C, whereas if any of the couplings is equal to zero, it is always possible to find such a representation that leads to a change in the sign of the field.
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